# American Institute of Mathematical Sciences

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## Nonautonomous gradient-like vector fields on the circle: Classification, structural stability and autonomization

 Institute of Information Technology, Mathematics and Mechanics, Lobachevsky National Research State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603950, Russia

* Corresponding author: lermanl@mm.unn.ru

To Jürgen, with the gratitude and best wishes

Received  December 2017 Revised  June 2018 Published  April 2019

We study a class of scalar differential equations on the circle $S^1$. This class is characterized mainly by the property that any solution of such an equation possesses an exponential dichotomy both on the semi-axes $\mathbb R_+$ and $\mathbb R_+$. Also we impose some other assumptions on the structure of the foliation into integral curves for such the equation. Differential equations of this class are called gradient-like ones. As a result, we describe the global behavior of a foliation, introduce a complete invariant of the uniform equivalency, give standard models for the equations of this distinguished class. The case of almost periodic gradient-like equations is also studied, their classification is presented.

Citation: Lev M. Lerman, Elena V. Gubina. Nonautonomous gradient-like vector fields on the circle: Classification, structural stability and autonomization. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020076
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##### References:
Bifurcation from infinities, $\mu = 0,$ $\mu > 0$
Bifurcation at the violation of Assumption 3
Foliation inside the rectangle
Neighborhoods: crosses correspond to u-solutions, bold points correspond to s-solutions
Construction of asymptotically autonomous ODE: crosses correspond to u-solutions, bold points correspond to s-solutions